3. Which of the following functions grow faster than x² as x→∞...

Question

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

c. √(x^4 + x^3)

Accepted Answer

Welcome back everyone. As x approaches infinity, which function, if any, grows faster? F of x equals square root of x 5 plus x 4, or g x equals x cubed. A F of X B G of X C, they grow at the same rate and d cannot be determined. For this problem, let's remember that to compare the growth weight of two functions. We want to find the limit as x approaches infinity of their ratio. We can take f of x in the numerator and g of x in the denominator. Substituting the functions, we get limit as x approaches infinity. Of the square root of x the power of 5 plus x the power of 4 divided by x cubed. Now, because X is positive, we can rewrite our denominator. S square root of x the power of 6, which returns that x cubed. And now in the numerator we have the square root of x the power of 5 plus x the power of 4. And this is particularly useful because now we can apply the properties of radicals. We can write everything under the same radicals, so that'd be square root of. X the power of 5 plus x the power of 4 divided by x the power of 6, and now we can perform division. We get the limit as x approaches infinity of square root of. X the power of 5 divided by x, the power of 6 gives us 1 divided by x to the power of 1, or basically 1 divided by x plus x to the power of 4 divided by x the power of 6 gives us 1 divided by x 2. And now as X approaches infinity. A constant divided by an infinitely large number is going to give us a limit of 0. So we get square root of 0 + 0, which is Sierra. Whenever we get a limit of zero, it basically means that the denominator function dominates over the numerator function, so g of x grows faster because the limit is 0 and g of x is in the denominator. So the correct answer for this problem corresponds to the answer choice B. Thank you for watching.

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?c. √(x^4 + x^3)

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3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. √(x^4 + x^3)